What is the general solution for 2cos((3x-pi)/6)+1=0,0<x<2pi ?

Q: What is the general solution for 2cos((3x-pi)/6)+1=0,0<x<2pi ?

The general solution for 2cos((3x-pi)/6)+1=0,0<x<2pi is x=(5pi)/3

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2cos((3x-pi)/6)+1=0,0<x<2pi

Solution

2 cos (\frac{3 x - π}{6}) + 1 = 0, 0 < x < 2 π

Solution

x = \frac{5 π}{3}

Degrees

x = 30 0^{\circ}

Solution steps

2 cos (\frac{3 x - π}{6}) + 1 = 0, 0 < x < 2 π

Move

1

to the right side

2 cos (\frac{3 x - π}{6}) + 1 = 0

Subtract

1

from both sides

2 cos (\frac{3 x - π}{6}) + 1 - 1 = 0 - 1

Simplify

2 cos (\frac{3 x - π}{6}) = - 1

2 cos (\frac{3 x - π}{6}) = - 1

Divide both sides by

2

2 cos (\frac{3 x - π}{6}) = - 1

Divide both sides by

2

\frac{2 cos ( \frac{3 x - π}{6} )}{2} = \frac{- 1}{2}

Simplify

cos (\frac{3 x - π}{6}) = - \frac{1}{2}

cos (\frac{3 x - π}{6}) = - \frac{1}{2}

General solutions for

cos (\frac{3 x - π}{6}) = - \frac{1}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{3 x - π}{6} = \frac{2 π}{3} + 2 πn, \frac{3 x - π}{6} = \frac{4 π}{3} + 2 πn

\frac{3 x - π}{6} = \frac{2 π}{3} + 2 πn, \frac{3 x - π}{6} = \frac{4 π}{3} + 2 πn

Solve

\frac{3 x - π}{6} = \frac{2 π}{3} + 2 πn : x = \frac{5 π}{3} + 4 πn

\frac{3 x - π}{6} = \frac{2 π}{3} + 2 πn

Multiply both sides by

6

\frac{3 x - π}{6} = \frac{2 π}{3} + 2 πn

Multiply both sides by

6

\frac{6 ( 3 x - π )}{6} = 6 \cdot \frac{2 π}{3} + 6 \cdot 2 πn

Simplify

\frac{6 ( 3 x - π )}{6} = 6 \cdot \frac{2 π}{3} + 6 \cdot 2 πn

Simplify

\frac{6 ( 3 x - π )}{6} : 3 x - π

\frac{6 ( 3 x - π )}{6}

Divide the numbers:

\frac{6}{6} = 1

= 3 x - π

Simplify

6 \cdot \frac{2 π}{3} + 6 \cdot 2 πn : 4 π + 12 πn

6 \cdot \frac{2 π}{3} + 6 \cdot 2 πn

6 \cdot \frac{2 π}{3} = 4 π

6 \cdot \frac{2 π}{3}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 π 6}{3}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{12 π}{3}

Divide the numbers:

\frac{12}{3} = 4

= 4 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 4 π + 12 πn

3 x - π = 4 π + 12 πn

3 x - π = 4 π + 12 πn

3 x - π = 4 π + 12 πn

Move

π

to the right side

3 x - π = 4 π + 12 πn

Add

π

to both sides

3 x - π + π = 4 π + 12 πn + π

Simplify

3 x = 5 π + 12 πn

3 x = 5 π + 12 πn

Divide both sides by

3

3 x = 5 π + 12 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{5 π}{3} + \frac{12 πn}{3}

Simplify

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

Solve

\frac{3 x - π}{6} = \frac{4 π}{3} + 2 πn : x = 3 π + 4 πn

\frac{3 x - π}{6} = \frac{4 π}{3} + 2 πn

Multiply both sides by

6

\frac{3 x - π}{6} = \frac{4 π}{3} + 2 πn

Multiply both sides by

6

\frac{6 ( 3 x - π )}{6} = 6 \cdot \frac{4 π}{3} + 6 \cdot 2 πn

Simplify

\frac{6 ( 3 x - π )}{6} = 6 \cdot \frac{4 π}{3} + 6 \cdot 2 πn

Simplify

\frac{6 ( 3 x - π )}{6} : 3 x - π

\frac{6 ( 3 x - π )}{6}

Divide the numbers:

\frac{6}{6} = 1

= 3 x - π

Simplify

6 \cdot \frac{4 π}{3} + 6 \cdot 2 πn : 8 π + 12 πn

6 \cdot \frac{4 π}{3} + 6 \cdot 2 πn

6 \cdot \frac{4 π}{3} = 8 π

6 \cdot \frac{4 π}{3}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{4 π 6}{3}

Multiply the numbers:

4 \cdot 6 = 24

= \frac{24 π}{3}

Divide the numbers:

\frac{24}{3} = 8

= 8 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 8 π + 12 πn

3 x - π = 8 π + 12 πn

3 x - π = 8 π + 12 πn

3 x - π = 8 π + 12 πn

Move

π

to the right side

3 x - π = 8 π + 12 πn

Add

π

to both sides

3 x - π + π = 8 π + 12 πn + π

Simplify

3 x = 9 π + 12 πn

3 x = 9 π + 12 πn

Divide both sides by

3

3 x = 9 π + 12 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{9 π}{3} + \frac{12 πn}{3}

Simplify

\frac{3 x}{3} = \frac{9 π}{3} + \frac{12 πn}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{9 π}{3} + \frac{12 πn}{3} : 3 π + 4 πn

\frac{9 π}{3} + \frac{12 πn}{3}

Divide the numbers:

\frac{9}{3} = 3

= 3 π + \frac{12 πn}{3}

Divide the numbers:

\frac{12}{3} = 4

= 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

x = \frac{5 π}{3} + 4 πn, x = 3 π + 4 πn

Solutions for the range

0 < x < 2 π

x = \frac{5 π}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos((3x-pi)/6)+1=0,0<x<2pi ?
The general solution for 2cos((3x-pi)/6)+1=0,0<x<2pi is x=(5pi)/3